Finite Group Databasebeta

← #30648977

Subgroups — #30648977

Order 16256.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550350C211yesmaximal, characteristic
2#22550214C212yesmaximal, complemented
3#22550173C212yesmaximal, complemented
4#22550078C212yesmaximal, complemented
5#22550026C212yesmaximal
6#22549974C212yesmaximal, complemented
7C4×#14523235C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic, complemented
9#541×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523235C2212yes—
12C2×#14523235C2212yes—
13C4×D508C2212yes—
14C4×D508C2212yes—
15#16861171—22no—
16#16861168C2211yescharacteristic
17C22⋊C4×D127—22no—
18#16861142C2212yescomplemented
19#16861140C2212yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21#16861133C2212yescomplemented
22#16861133C2212yescomplemented
23#16861129C2212yes—
24#16861129C2212yes—
25C22⋊(C127⋊C8)C2212yes—
26#16861120C2211yescharacteristic, complemented
27C4×C127⋊C8C2211yescharacteristic, complemented
28C4⋊Q8×C127C2211yescharacteristic, complemented
29#126×C127—22no—
30#125×C127—22no—
31#125×C127C2211yescharacteristic, complemented
32C22⋊Q8×C127C2212yes—
33C4×Q8×C127C2212yescomplemented
34#117×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—24no—
37C2×D508—21no—
38C2×D508C2312yes—
39C2×C4×D127—22no—
40C2×C4×D127—22no—
41#14523235—24no—
42#14523235—24no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127C2312yes—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—22no—
48(C2×D127)⋊C4—22no—
49C4×Dic127—21no—
50C2×Q8×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127—22no—
56C4⋊C4×C127D412yes—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—22no—
59C22⋊C4×C127D412yes—
60C42×C127—22no—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—82no—
66D508—24no—
67D508—22no—
68D508#4114yes—
69C2×Dic127—22no—
70C4×D127—82no—
71C4×D127—82no—
72C127⋊C8—22no—
73C127⋊C8—22no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—22no—
76Q8×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2411yescharacteristic
84C22×C127—21no—
85C22×C127—22no—
86C22×C127C22×D411yescharacteristic
87C2×D127—24no—
88C2×D127—42no—
89C2×D127—42no—
90C2×D127—84no—
91C4×C127—21no—
92C4×C127—22no—
93C4×C127—22no—
94C4×C127—42no—
95C4×C127—42no—
96C4×C127C2×#4112yes—
97Dic127—42no—
98C2×C127—41no—
99C2×C127#52511yescharacteristic
100C2×C127#57712yes—
101D127—44no—
102#3123—1271nomaximal, Sylow
103C127#312311yescharacteristic, complemented, Sylow
104#541D12711yescharacteristic, complemented
105#518—1271no—
106#485—1272no—
107#478—1272no—
108#461—1272no—
109#450—1272no—
110#440—1272no—
111C4×SD16—1272no—
112#433—1271no—
113C22×D4—1271no—
114C2×SD16—1272no—
115C2×SD16—1272no—
116C4⋊Q8C2×D12711yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D12711yescharacteristic, complemented
120C22⋊Q8C2×D12712yes—
121C4×Q8C2×D12712yescomplemented
122C4×D4—1272no—
123C4×D4—1272no—
124C4×D4—2542no—
125#117—1271no—
126#117C2×D12711yescharacteristic
127C2×C22⋊C4—2542no—
128#107—1272no—
129#106—1272no—
130C4⋊C8—1272no—
131#103—1272no—
132#103—1272no—
133#102—1272no—
134#102—1272no—
135C22⋊C8—1272no—
136#97—1271no—
137C4×C8—1271no—
138C24—2541no—
139C2×Q8C22×D12712yes—
140C2×D4—1272no—
141C2×D4—2544no—
142C2×D4—2541no—
143C22×C4—2542no—
144C22×C4—2542no—
145C22×C4C22×D12711yescharacteristic
146SD16—2544no—
147SD16—2544no—
148C2×C8—1272no—
149C2×C8—1272no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—1272no—
155C4⋊C4C22⋊D12712yes—
156C4⋊C4C22×D12712yes—
157C22⋊C4—22no—
158C22⋊C4—2541no—
159C22⋊C4—2542no—
160C22⋊C4—2542no—
161C22⋊C4C22⋊D12712yes—
162C42—22no—
163C42—2541no—
164C42C22×D12711yescharacteristic
165C42C22×D12711yescharacteristic
166C23—10162no—
167C23—2544no—
168C23—2542no—
169C23C2×C22⋊D12711yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—1274no—
173D4—2544no—
174D4—2542no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—10162no—
179C2×C4—10162no—
180C2×C4—2542no—
181C2×C4C2×C22⋊D12712yes—
182C2×C4C2×C22⋊D12711yescharacteristic
183C2×C4C2×C22⋊D12712yes—
184C2×C4C23×D12711yescharacteristic
185C8—2542no—
186C8—2542no—
187C22—21no—
188C22—22no—
189C22—10164no—
190C22—5082no—
191C22—5082no—
192C22—2544no—
193C22C22×C22⋊D12711yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—5082no—
200C4#41×D12712yes—
201C2—41no—
202C2—5084no—
203C2#2255038311yescharacteristic
204C2#2255068412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#541×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#541D127
hypocenterC127#3123
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22550214C2semidirect
3. #22550173C2semidirect
4. #22550078C2semidirect
6. #22549974C2semidirect
7. C4×#14523235C2semidirect
8. #22549937C2semidirect
9. #541×C127C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
21. #16861133C22semidirect
22. #16861133C22semidirect
26. #16861120C22semidirect
27. C4×C127⋊C8C22semidirect
28. C4⋊Q8×C127C22semidirect
31. #125×C127C22semidirect
33. C4×Q8×C127C22semidirect
103. C127#3123semidirect
104. #541D127semidirect
116. C4⋊Q8C2×D127semidirect
119. #125C2×D127semidirect
121. C4×Q8C2×D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225503502
#225502142
#225501732
#225500782
#225500262
#225499742
C4×#145232352
#225499372
#541×C1272
#3123127